Lotka's law, named after Alfred J. Lotka, is one of a variety of special applications of Zipf's law. It describes the frequency of publication by authors in any given field. Let X be the number of publications, Y be the number of authors with X publications, and k be a constants depending on the specific field. Lotka's law states that. In Lotka's original publication, he claimed k=2. Subsequent research showed that k varies depending on the discipline. Equivalently, Lotka's law can be stated as, where Y' is the number of authors with at least X publications. Their equivalence can be proved by taking the derivative.

Example

Assume that n=2 in a discipline, then as the number of articles published increases, authors producing that many publications become less frequent. There are 1/4 as many authors publishing two articles within a specified time period as there are single-publication authors, 1/9 as many publishing three articles, 1/16 as many publishing four articles, etc. And if 100 authors wrote exactly one article each over a specific period in the discipline, then: That would be a total of 294 articles and 155 writers, with an average of 1.9 articles for each writer.

Software

Relationship to Riemann Zeta

Lotka's law may be described using the Zeta distribution: for and where is the Riemann zeta function. It is the limiting case of Zipf's law where an individual's maximum number of publications is infinite.